Class 12 RD Sharma Solutions – Chapter 25 Vector or Cross Product – Exercise 25.1 | Set 2

Question 13. If , and , find 

Solution:

We know that,

=> 

=>

=> 

=> 

=>

Also,

=>

And 

=> 

=> 

=> 

=> 

=> 

=>

Question 14. Find the angle between 2 vectors and , if 

Solution:

Given 

=>

=>, as  is a unit vector.

=> 

=> 

=> 

Question 15. If , then show that , where m is any scalar.

Solution:

Given that 

=> 

=> 

=>

Using distributive property,

=> 

If two vectors are parallel, then their cross-product is 0 vector.

=>  and  are parallel vectors.

=> 

Hence proved.

 Question 16. If, and , find the angle between  and 

Solution:

Given that,, and 

We know that,

=> 

=> 

=> 

=> 

=> 

=> 

=> 

=> 

Question 17. What inference can you draw if and 

Solution:

Given, and 

=>

=>

Either of the following conditions is true,

1.

2. 

3. 

4.is parallel to 

=> 

=> 

Either of the following conditions is true,

1. 

2.

3. 

4. is perpendicular to 

Since both these conditions are true, that implies atleast one of the following conditions is true,

1. 

2. 

3. 

Question 18. If , and are 3 unit vectors such that , and . Show that ,and  form an orthogonal right handed triad of unit vectors.

Solution:

Given, , and 

As,

=>

=>  is perpendicular to both and .

Similarly,

=>  is perpendicular to both and 

=>  is perpendicular to both and 

=> , and  are mutually perpendicular.

As, , and  are also unit vectors,

=> , and  form an orthogonal right-handed triad of unit vectors

Hence proved.

Question 19. Find a unit vector perpendicular to the plane ABC, where the coordinates of A, B, and C are A(3, -1, 2), B(1, -1, 3), and C(4, -3, 1).

Solution:

Given A(3, -1, 2), B(1, -1, 3) and C(4, -3, 1).

Let,

=> 

=> 

=> 

Plane ABC has two vectors  and 

=> 

=> 

=> 

=> 

=> 

=> 

=> 

=> 

A vector perpendicular to both and is given by,

=> 

=> 

=> 

=> 

=>

To find the unit vector,

=> 

=> 

=> 

=> 

Question 20. If a, b and c are the lengths of sides BC, CA and AB of a triangle ABC, prove that and deduce that 

Solution:

Given that , and 

From triangle law of vector addition, we have

=> 

=> 

=> 

=> 

=> 

=> 

=> 

=> 

=> 

=> 

=> 

=> 

=>

=> 

=> 

Similarly,

=> 

=> 

=> 

=> 

=> 

=> 

=>

=> 

=> 

=> 

=> 

Hence proved.

Question 21. If and , then find . Verify that and  are perpendicular to each other.

Solution:

Given, and 

=>

=>

=> 

=>

=> 

Two vectors are perpendicular if their dot product is zero.

=>

=> 

=> 

=> 

Hence proved.

Question 22. If  and  are unit vectors forming an angle of , find the area of the parallelogram having and  as its diagonals.

Solution:

Given and forming an angle of .

Area of a parallelogram having diagonals  and  is 

=> 

=> 

=>

Thus area is,

=> Area = 

=> Area = 

=> Area = 

=> Area = 

=> Area = 

=> Area =

=> Area = 

=> Area = 

=> Area =  square units

Question 23. For any two vectors  and  , prove that 

Solution:

We know that,

=> 

=>

=>

=>

=> 

=> 

=> 

=>

=>

Hence proved.

Question 24. Define and prove that , where  is the angle between and 

Solution:

Definition of : Let  and be 2 non-zero, non-parallel vectors. Then , is defined as a vector with the magnitude of , and which is perpendicular to both the vectors  and .

We know that,

=> 

=> 

=> ……………..(eq.1)

And as,

=> 

=>

Substituting in (eq.1),

=> 

=>